Markov's theorem

The set of Markov is a theorem from the mathematical field of knot theory , it is sufficient and necessary conditions , when the statements of two braids equivalent tangles arise.

Markov trains

Type I Markov train

Type II Markov train

The picture on the right shows the two types of Markov trains, each of which converts one Zopf diagram into another.

Markov train of type I (conjugation): In the braid group , this train corresponds to conjugation with one word , a braid is therefore transferred to the braid . ${\ displaystyle B_ {n}}$ ${\ displaystyle b \ in B_ {n}}$ ${\ displaystyle from}$ ${\ displaystyle ba}$

Markov train of type II (stabilization): In the braid group , this train corresponds to the multiplication of an element from with the producer or its inverse . The reverse operation is called destabilization.

{\ displaystyle B_ {n}}

{\ displaystyle B_ {n-1} \ subset B_ {n}}

{\ displaystyle \ sigma _ {n-1}}

{\ displaystyle \ sigma _ {n-1} ^ {- 1}}

Markov's theorem

The ends of two braids are equivalent entanglements if and only if the corresponding elements of the braid group can be converted into one another through a series of conjugations and stabilizations / destabilizations. (Equivalent: if the corresponding Zopf diagrams can be converted into one another by a sequence of Markov trains and isotopic of Zopf diagrams.)

Applications

Quantum invariants of knots and links are defined using a representation of the link as the end of a braid. In order to prove the well-definedness of the node invariants, the invariance of the respective invariant has to be checked under Markov trains.

literature

AA Markov : On the free equivalence of closed braids. Recueil Mathématique Moscou, 1 (1935), pp. 73-78.
NM Weinberg: On free equivalence of free braids. CR (Doc.) Acad. Sci. USSR, 23 (1939) pp. 215-216. (Russian)
Joan Birman : Braids, Links and Mapping Class Groups. Annals of Math. Studies 82 (1974).
H. Morton: Threading knot diagrams. Math Proc. Camb. Phil. Soc. 99: 247-260 (1986).
S. Lambropoulou, C. Rourke: Markov's theorem in 3-manifolds. Topology and its Applications 78, Nos. 1-2 (1997), pp. 95-122.
P. Traczyk: A new proof of Markov's braid theorem. Knot Theory, Banach Center Publications 42, Polish Acad. of Sciences (1998), pp. 409-419.
Joan Birman, William Menasco: On Markov's theorem. Knots 2000 Korea, Vol. 1 (Yongpyong). J. Knot Theory Ramifications 11 (2002), no. 3, pp. 295-310 ( online ).

Web links

Markov Moves (MathWorld)
Markov's Theorem (MathWorld)
Markov braid theorem (Encyclopedia of Mathematics)